Abstract: Abstract Let \((X,\,D)\) be an m-pointed compact Riemann surface of genus at least 2. For each \(x \,\in \, D\) , fix full flag and concentrated weight system \(\alpha \) . Let \(P \mathcal {M}_{\xi }\) denote the moduli space of semi-stable parabolic vector bundles of rank r and determinant \(\xi \) over X with weight system \(\alpha \) , where r is a prime number and \(\xi \) is a holomorphic line bundle over X of degree d which is not a multiple of r. We compute the Chen–Ruan cohomology of the orbifold for the action on \(P \mathcal {M}_{\xi }\) of the group of r-torsion points in \(\mathrm{Pic}^0(X)\) . PubDate: 2022-05-17
Abstract: Abstract Let X be a tree of proper geodesic spaces with edge spaces strongly contracting and uniformly separated from each other by a number depending on the contraction function of edge spaces. Then we prove that the strongly contracting geodesics in vertex spaces are quasiconvex in X. We further prove that in X if all the vertex spaces are uniformly hyperbolic metric spaces, then X is a hyperbolic metric space and vertex spaces are quasiconvex in X. PubDate: 2022-05-11
Abstract: Abstract In this paper, we are interested in the fractional Yamabe-type equation \(A_s u= u^\frac{n+2s}{n-2s}\) , \({u>0} \text{ in } \Omega \text{ and } u=0 \text{ on } \partial \Omega .\) Here \(\Omega \) is a regular bounded domain of \({\mathbb {R}}^n, n\ge 2\) and \(A_s, s\in (0, 1)\) represents the fractional Laplacian operator in \(\Omega \) with zero Dirichlet boundary condition. Based on the theory of critical points at infinity of Bahri and the localization technique of Caffarelli and Silvestre, we compute the difference of topology induced by the critical points at infinity between the level sets of the variational functional associated to the problem. Our result can be seen as a nonlocal analog of the theorem of Bahri et al. (Cal. Var. Partial. Differ. Equ. 3 (1995) 67–94) on the classical Yamabe-type equation. PubDate: 2022-05-11
Abstract: Abstract Let G be a group with identity element e. The proper power graph and proper enhanced power graph of G, are denoted by \(\Gamma ^*_{P}(G)\) and \(\Gamma ^*_{EP}(G)\) , respectively. Also, the prime graph of G is denoted by \(\Gamma _{GK}(G)\) . In an article, Aalipour et al. (Electronic J. Combin. 24(3) (2017) 3–16) asked which groups do have the property that \(\Gamma ^*_{P}(G)\) is connected' In this paper, we show that if \(\Gamma _{GK}(G)\) is disconnected, then \(\Gamma ^*_{P}(G)\) and \(\Gamma ^*_{EP}(G)\) are disconnected. Moreover, we prove that if G is a nilpotent group which is not a p-group, then \(\Gamma ^*_{EP}(G)\) is a connected graph. PubDate: 2022-05-07
Abstract: Abstract In this paper, we answer two long-standing questions on the classification of G-torsors on curves for an almost simple, simply connected algebraic group G over the field of complex numbers. The first is the construction of a flat degeneration of the moduli of G-torsors on smooth projective curves when the smooth curve degenerates to an irreducible nodal curve and the second one is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible projective nodal curve. A generalization of the classical Bruhat–Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools. PubDate: 2022-05-07
Abstract: Abstract A result of Hurwitz says that the special linear group of size greater than or equal to three over the ring of integers of an algebraic number field is finitely generated. A celebrated theorem in group theory states that finite simple groups are two-generated. Since the special linear group of size at least three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In this paper, we provide a set of three generators for the special linear group of size three over the rings of integers of imaginary quadratic number fields of class number one. The speciality of this set of generators is that it is unbiased towards the choice of a particular simple root (from a Lie algebra point of view). This new set of generators is inspired by the work of Gow and Tamburini for the special linear group over the (ring of rational) integers. PubDate: 2022-04-21
Abstract: Abstract Let X be a K3 surface and L be an ample line bundle on it. In this article, we will give an alternative and elementary proof of Lelli-Chiesa’s theorem in the case of \(r= 2\) . More precisely, we will prove that under certain conditions the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system L . Using Lelli-Chiesa’s theorem for \(r \ge 3\) , we also extend Lelli-Chiesa’s theorem in the case of \(r= 2\) in weaker condition. PubDate: 2022-04-20
Abstract: Abstract We give an exact criterion of a conjecture of L. M. Kelly to hold true which is stated as follows. If there is a finite family \(\Sigma \) of mutually skew lines in \(\mathbb {R}^d,d\ge 4\) such that the 3-flat spanned by every two lines in \(\Sigma \) , contains at least one more line of \(\Sigma \) , then we have that all of the lines of \( \Sigma \) are contained in a single 3-flat if and only if the arrangement of 3-flats is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, where we prove that, for (2, 5)-representations of Sylvester–Gallai designs in \(\mathbb {R}^6\) , the analogous statement does not hold. PubDate: 2022-04-20
Abstract: Abstract Let \(X \subset {\mathbb {P}}^3\) be a very general sextic surface over complex numbers. In this paper, we study certain Brill–Noether problems for moduli of rank 2 stable bundles on X and its relation with rank 2 weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill–Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank 2 weakly Ulrich bundles \({\mathcal {E}}\) on X with \(c_1({\mathcal {E}}) =5H\) and \(c_2 \ge 91\) and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill–Noether loci. Finally, using the existence of rank 2 Ulrich bundles on X we obtain some more non-empty Brill–Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on X. PubDate: 2022-04-19
Abstract: Abstract An orbit pattern \(\alpha \) is said to force an orbit pattern \(\beta \) , if any continuous interval map which admits \(\alpha \) also admits \(\beta \) . Among the orbit patterns that force only eventually fixed trajectories, we completely describe the forcing relation, by answering the question: which orbit patterns force which others' We provide two different ways to enlist them completely through formal languages. One is through constructed words and another is by derived words. PubDate: 2022-04-19
Abstract: Abstract In this article, new results on the Gabriel localizations are obtained. As an application of them, it is shown that a morphism of rings is a flat epimorphism of rings if and only if it corresponds to a kind of the Gabriel localizations. Using this result, new progress in the understanding of the structure of flat epimorphisms of rings have been made. Especially among them, a set-theoretical gap in the structure of the ring M(R), the maximal flat epimorphic extension of a ring R, has been fixed. PubDate: 2022-04-16
Abstract: Abstract In this paper, we show that the solution of an infinite delay equations in a Banach space given by non-linear semigroups in Fréchet spaces can be approximated by finite delay equations. PubDate: 2022-04-16
Abstract: Abstract Let L be a one-to-one operator of type w, with \(w\in [0,\pi /2]\) , satisfying the Davies–Gaffney estimates. For \(\alpha \in (0,\infty )\) and \(p\in (0,\infty )\) and under the condition that \(q(\cdot ): {{{\mathbb {R}}}}^{n}\longrightarrow [1,\infty )\) satisfies the globally log-Hölder continuity condition, we introduce the Herz-type Hardy space with variable exponents associated to L and establish its molecular decomposition. The atomic characterization and maximal function characterizations of the space are proved under the assumption that L is a non-negative self-adjoint operator on \(L^{2}({{{\mathbb {R}}}}^{n})\) whose heat kernels satisfy the Gaussian upper bound estimates. All the results are new even for the constant case. PubDate: 2022-04-13
Abstract: Abstract In this article, we prove the transcendence of special values of some Hurwitz zeta type series. Moreover, we find a linear independence criterion of these series under some mild conditions. We also show that, for any positive integer k and for any \( a, b \in (0, 1) \cap {\mathbb {Q}}\) with \( a+b = 1\) , at least one of the \(\zeta (2k, a) \) or \(\zeta (2k, b) \) must be transcendental. PubDate: 2022-03-28
Abstract: Abstract We study ring structure of the big ordinary Hecke algebra \({{\mathbb {T}}}\) with the modular deformation \(\rho _{{\mathbb {T}}}:{\mathrm {Gal}}({\bar{{{\mathbb {Q}}}}}/{{\mathbb {Q}}})\rightarrow {\mathrm {GL}}_2({{\mathbb {T}}})\) of an induced Artin representation \({\text {Ind}}_F^{{\mathbb {Q}}}\varphi \) from a real quadratic field F with a fundamental unit \(\varepsilon \) , varying a prime \(p\ge 3\) split in F. Under mild assumptions (H0)–(H3) given in the Introduction (on the prime p), we prove that \({{\mathbb {T}}}\) is an integral domain free of even rank \(e>0\) over \(\Lambda \) for the weight Iwasawa algebra \(\Lambda \) étale outside \({\mathrm {Spec}}(\Lambda /p(\langle \varepsilon \rangle -1))\) for \(\langle \varepsilon \rangle {:}{=}(1+T)^{\log _p(\varepsilon )/\log _p(1+p)}\in {{\mathbb {Z}}}_p[[T]]\subset \Lambda \) . If \(p\not \mid e\) , \({{\mathbb {T}}}\) is shown to be a normal noetherian domain of dimension 2 with ramification locus exactly given by \((\langle \varepsilon \rangle -1)\) . Moreover, only under p-distinguishedness (H0), we prove that any modular specialization of weight \(\ge 2\) of \(\rho _{{\mathbb {T}}}\) is indecomposable over the inertia group at p (solving a conjecture of Greenberg without exception). PubDate: 2022-03-28
Abstract: Abstract Let G be a finite abelian group and \(A\subset \mathbb Z\) . The weighted zero-sum constant \(s_A(G)\) (resp. \(\eta _A(G)\) ) is defined as the least positive integer t, such that every sequence S over G with length \(\ge t\) has an A-weighted zero-sum subsequence of length \(\mathrm{exp}(G)\) (resp. \(\le \!\!\exp (G)\) ). In this article, we investigate the value of \(s_A(G)\) and \(\eta _A(G)\) in the case \(G=\mathbb Z_n\oplus \mathbb Z_n\oplus \cdots \oplus \mathbb Z_n\) , where n is a square-free odd integer and A is the set of integers co-prime to n. We also obtain certain properties about extremal zero-sum free sequences. PubDate: 2022-03-22
Abstract: Abstract The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let \(\{X_k,k\ge 1\}\) be a sequence of independent and identically distributed random variables. Under a fairly general condition, an universal result in almost sure local limit theorem for the partial sums \(S_k=\sum \nolimits _{i=1}^kX_i\) is established on the weight \(d_k=k^{-1}\exp (\log ^\beta k)\) , \(0\le \beta <1/2\) : $$\begin{aligned}&\underset{n\rightarrow \infty }{\lim } \frac{1}{D_n} \sum \limits _{k=1}^nd_k\frac{\mathrm{I}(a_k\le S_k<b_k)}{\mathrm{P}(a_k\le S_k <b_k)}=1 \ \ \ \mathrm a.s., \end{aligned}$$ where \(D_n=\sum \nolimits _{k=1}^nd_k\) , \(-\infty \le a_k\le 0 \le b_k \le \infty ,\ \ \ k=1,2,\ldots \) . This result extends previous results in the almost sure local central limit theorems from \(d_k=1/k\) to \(d_k=k^{-1}\exp (\log ^\beta k)\) , \(0\le \beta <1/2\) . PubDate: 2022-03-22
Abstract: Abstract In this paper, we explicitly compute the derivation module of quotients of polynomial rings by ideals formed by the sum or by some other gluing technique. We discuss cases of monomial ideals and binomial ideals separately. PubDate: 2022-03-22
Abstract: Abstract We show that an analog of the Furstenberg–Zimmer structure theorem holds for \(\sigma \) -finite non atomic measure spaces and measure preserving strongly recurrent actions of discrete groups. We adapt the idea of Tao in associating Hilbert modules to measure preserving extensions and show that for an isomorphic copy of the \(L^2\) -space, the tools of Zimmer structure theory could be applied. PubDate: 2022-03-22
Abstract: Abstract Let \(k \,=\, \mathbb {Q}(\root 5 \of {n},\zeta _5)\) , where n is a positive integer 5-th power-free, whose 5-class group denoted by \(C_{k,5}\) is isomorphic to \(\mathbb {Z}/5\mathbb {Z}\times \mathbb {Z}/5\mathbb {Z}\) . Let \(k_0\,=\,\mathbb {Q}(\zeta _5)\) be the cyclotomic field containing a primitive 5-th root of unity \(\zeta _5\) . Let \(C_{k,5}^{(\sigma )}\) be the group of ambiguous classes under the action of \(\mathrm{Gal}(k/k_0)\) = \(\langle \sigma \rangle \) . The aim of this paper is to determine all naturals n such that the group of ambiguous classes \(C_{k,5}^{(\sigma )}\) has rank 1 or 2. PubDate: 2022-03-18
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